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A conjecture appears in \cite{milsteinscheme}, in the form of a remark, whereit is stated that it is possible to construct, in a specified way, any highorder explicit numerical schemes to approximate the solutions of SDEs withsuperlinear coefficients. We answer this conjecture affirmatively for the caseof order 1.5 approximations and show that the suggested methodology works.Moreover, we explore the case of having H\"{o}lder continuous derivatives forthe diffusion coefficients.

On explicit order 1.5 approximations with varying coefficients: the case of super-linear diffusion coefficients